Systems and methods for real time kinematic satellite positioning

ABSTRACT

A method for Real Time Kinematic satellite positioning includes receiving navigation satellite carrier signals, receiving phase correction signals from a reference station, calculating a set of integer phase ambiguities from double-differenced measurements of pseudo-range and phase, and calculating a relative position of the mobile receiver from the set of integer phase ambiguities and the double-differenced measurements of pseudo-range and phase.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 62/069,153, filed on 27 Oct. 2014, which is incorporated in itsentirety by this reference.

TECHNICAL FIELD

This invention relates generally to the satellite-based positioningfield, and more specifically to new and useful systems and methods forReal Time Kinematic (RTK) satellite positioning in the satellite-basedpositioning field.

BACKGROUND

Being able to perform high precision satellite positioning is importantfor a wide variety of applications. Unfortunately, current GPS solutionsare often either inaccurate or require processor power beyond thecapabilities of inexpensive mobile hardware. Thus, there is the need inthe satellite-based positioning field to create systems and methods forReal Time Kinematic (RTK) satellite positioning. This invention providessuch new and useful systems and methods.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a diagram view of a system of a preferred embodiment;

FIG. 2 is a diagram view of RTK signal reception;

FIG. 3 is a chart view of a mobile receiver of a system of a preferredembodiment;

FIG. 4 is a chart view of a method of a preferred embodiment; and

FIG. 5 is an example view of a transformation of a method of a preferredembodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The following description of the preferred embodiments of the inventionis not intended to limit the invention to these preferred embodiments,but rather to enable any person skilled in the art to make and use thisinvention.

1. RTK Satellite Positioning System

As shown in FIG. 1, a Real Time Kinematic (RTK) satellite positioningsystem 100 includes one or more reference stations 110 and a mobilereceiver 120. The system 120 may additionally include a centralprocessing server 130.

The system 100 functions to estimate the position of the mobile receiver120 with high accuracy using RTK satellite navigation. Typical satellitepositioning systems (e.g., standard GNSS) work by attempting to align alocal copy (at a receiver) of a pseudorandom binary sequence with asatellite-transmitted copy of the same sequence; because the satelliteis far from the receiver, the signal transmitted by the satellite isdelayed. By delaying the local copy of the sequence to match up with thesatellite-transmitted copy, the time it takes the signal to travel fromthe satellite to the receiver can be found, which can in turn be used tocalculate the distance between the satellite and receiver. By performingthis process for multiple satellites (typically four or more), aposition of the receiver relative to the satellites can be found, whichcan in turn be used to find the position in a particular geographiccoordinate system (e.g., latitude, longitude, and elevation). TypicalGNSS systems can achieve at best 2 m accuracy in positioning.

Instead of solely using the content of satellite signals, RTK makes useof satellite signal carriers to determine position. Both a referencestation receiver and a mobile receiver measure the phase of a receivedcarrier signal, as shown in FIG. 2. If the difference in absolute phasebetween the signal received at the reference station (also referred toas a base station) and the signal received at the mobile receiver can befound, the position of the mobile receiver relative to the referencestation can be found (this relative position may be represented as thevector b). Note that it may be difficult to determine difference inabsolute phase—because the carrier signal is uniform, it may not bepossible to differentiate between a phase shift of φ and 2πN+φ usingphase measurements alone, where N is an integer. For example, it may bedifficult to determine the difference between a phase shift of π radiansand a phase shift of 3π radians (or −π, 5π, etc.). This problem is knownas integer ambiguity.

If the relative position can be found and the position of the referencestation is known with high accuracy, the position of the mobile receivercan be found with accuracy on the order of centimeters (by referencingthe RTK-determined relative position to the position of the referencestation). Limits on positioning accuracy are preferably limiting factorsin determination of relative position (i.e., the reference position isknown with higher accuracy/precision than the relative position);alternatively, it may be more important to determine relative position(i.e., relative position is known with higher accuracy/precision thanthe reference position).

The system 100 functions to perform satellite positioning using a novelform of RTK (described in the sections covering the method 200). Mobilereceivers 120 of the system 100 are preferably able to achievecentimeter-level relative positioning while maintaining a small formfactor, low cost, and low power consumption. The advantages in accuracy,size, cost, and power consumption preferably enable mobile receivers 120to be used in applications where previous GNSS solutions were notaccurate enough, too large, too expensive, and/or too power hungry.Applications that may be of particular interest to purchasers of mobilereceivers 120 include autonomous vehicle guidance (e.g., for UAVs oragricultural equipment), GPS/GNSS research, and surveying systems.Additionally, the mobile receivers 120 are preferably designed toutilize open-source firmware, allowing them to be easily customized toparticular demands of end user applications, easing system integrationand reducing host system overhead.

The reference stations 110 function to transmit phase data of signalsreceived at the reference stations 110. The reference stations noutilized by the system 100 are preferably public reference stations, butmay additionally or alternatively be private reference stations or anyother suitable reference stations.

Reference stations 110 preferably have a location known to a high degreeof accuracy. Reference station 110 location is preferably the locationof the antenna used to receive satellite signals. Reference station 110location may be determined in any manner yielding a high degree ofaccuracy; for example, reference station 110 location may be determinedby a number of single frequency carrier phase receivers set around thereference station no at vertical and horizontal reference points. Notethat while reference stations no are preferably fixed in location, theymay additionally or alternatively be mobile. Station position ispreferably re-determined to high accuracy before moved referencestations no re-start providing phase data; additionally oralternatively, reference stations no may provide phase data beforelocation re-determination (for example, for use in attitude estimation).As another alternative, reference stations no may not provide absolutelocation data at all if not needed; for example, absolute location dataof the reference station no may not be needed for applications includingattitude estimation.

Reference stations 110 preferably provide phase data for multiplesatellite signals and the location of the reference station via theinternet, but may additionally or alternatively provide data by anyother suitable method (e.g., transmission by UHF-band radio modem).Reference station no data is preferably made available directly tomobile receivers 120, but may additionally or alternatively be processedor aggregated before being made available to mobile receivers 120.

In one variation of a preferred embodiment, data from multiple referencestations 110 is combined at a server (e.g., the central processingserver 130); the server uses the reference station no data to create avirtual reference station. Error in relative positioning of a mobilereceiver 120 increases with the distance from the reference station no.By comparing data from multiple reference stations 110,distance-dependent systematic errors (e.g., those caused by ionosphericand tropospheric refractions or satellite orbit errors) can be modeledmore precisely. The server can then use these error models to predictthe reference data that would be transmitted by a reference station nearthe mobile receiver 120; from this prediction, data from a ‘virtualreference station’ with a location near the mobile receiver 120 can betransmitted to the mobile receiver 120 and used to increase mobilereceiver 120 positioning accuracy.

The mobile receiver 120 functions to calculate a position relative toone or more reference stations 110 of known location using signalstransmitted from one or more positioning satellites (preferably at least3, but alternatively any number) and reference data from the referencestations no. The mobile receiver 120 preferably calculates position datausing the method 200, but may additionally or alternatively calculateposition data in any suitable manner. Mobile receivers 120 mayadditionally or alternatively serve as reference stations no for othermobile receivers 120.

As shown in FIG. 3, the mobile receiver 120 preferably includes anantenna coupler 121, a front-end module 122, a satellite signalmanagement module 123, a microcontroller 124, and an input/output module125.

The antenna coupler 121 functions to couple a satellite signal-readyantenna to the mobile receiver 120 (additionally or alternatively, themobile receiver 120 may include an antenna, such as a patch antenna).

Antennas coupled by and/or included with the antenna coupler 121 arepreferably made out of a conductive material (e.g., metal). The antennasmay additionally or alternatively include dielectric materials to modifythe properties of the antennas or to provide mechanical support.

The antennas may be of a variety of antenna types; for example, patchantennas (including rectangular and planar inverted F), reflectorantennas, wire antennas (including dipole antennas), bow-tie antennas,aperture antennas, loop-inductor antennas, and fractal antennas. Theantennas can additionally include one or more type of antennas, and thetypes of antennas can include any suitable variations. The antennastructure may be static or dynamic (e.g., a wire antenna that includesmultiple sections that may be electrically connected or isolateddepending on the state of the antenna). Antennas may have isotropic oranisotropic radiation patterns (i.e., the antennas may be directional).If antennas are directional, their radiation pattern may be dynamicallyalterable; for example, an antenna substantially emitting radiation inone direction may be rotated so as to change the direction of radiation.

If the antenna coupler 121 couples to multiple antennas, the antennacoupler 121 may split power between them using a splitter; additionallyor alternatively, the antenna coupler 121 may include a switch to selectbetween the multiple antennas, or the antenna coupler 121 may couple tothe antennas in any suitable manner.

The front-end module 122 functions to convert signals received by theantenna coupler 121 to digital baseband signals for processing. Thefront-end module 122 includes an analog-to-digital converter (e.g., theMaxim MAX2769) capable of operating at high sample rates. The front-endmodule 122 is preferably capable of receiving L1 GPS, GLONASS, Galileo,and SBAS signal bands. The front-end module 122 may additionally oralternatively be capable of receiving additional bands (e.g., L2 GPS) orthe mobile receiver 120 may include multiple front-end modules 122 fordifferent bands.

The satellite signal management module 123 functions to performsatellite signal tracking and acquisition. The satellite signalmanagement module 123 may additionally or alternatively includeprogrammable digital notch filters for performing continuous wave noisenulling. The satellite signal management module preferably includesflexible and fully programmable correlators that may be used by themicrocontroller 124 to implement tracking loops and acquisitionalgorithms. The satellite signal management module 123 is preferablyimplemented on an FPGA, allowing the firmware to be altered to enableadaptation of the mobile receiver 120 to various applications.Additionally or alternatively, the satellite signal management module123 may be implemented by any suitable circuit.

The microcontroller 124 functions to perform signal processing above thecorrelator level on the mobile receiver 120 (e.g., tracking loopfilters, acquisition management, navigation processing, etc.). Themicrocontroller 124 additionally or alternately manages communicationover the input/output module 125. The microcontroller 124 preferably isable to calculate position, velocity, time (PVT) solutions at a rate of50 Hz or higher, but may additionally or alternatively calculate PVTsolutions at any suitable rate.

The input/output module 125 functions to allow for data to betransmitted from or received by the mobile receiver 120. Theinput/output module 125 is preferably used to couple the mobile receiver120 to a UHF radio modem, so that the mobile receiver 120 may receivereference station correction signals over the UHF radio modem. Theinput/output module 125 may additionally or alternatively be used forany other suitable transmission or reception of data from the mobilereceiver 120 (e.g., the mobile receiver 120 may transmit raw navigationdata over the input/output module 125 to a control computer on a UAV, orthe mobile receiver 120 may transmit data through a Bluetooth orcellular modem connected to the input/output module 125). Theinput/output module 125 preferably includes one or more UARTconnections, but may additionally or alternatively include connectionsfor any other suitable input/output communications; for example, theinput/output module 125 may include a USB port.

The central processing server 130 functions to process data fromreference stations no and mobile receivers 120. The central processingserver 130 may process this data for multiple purposes, includingaggregating position data (e.g., tracking multiple mobile receivers120), system control (e.g., providing flight directions to a UAV basedon position data received from a mobile receiver 120 attached to theUAV), and/or position calculation (e.g., performing calculations formobile receivers 120 that are offloaded due to limited memory orprocessing power). The central processing server 130 may additionally oralternatively process data to perform integer ambiguity calculations,used in determining RTK position solutions. The central processingserver 130 may additionally or alternatively manage reference stationsno or generate virtual reference stations for mobile receiver 120 basedon reference station 110 data. The central processing server 130 mayadditionally or alternatively serve as an internet gateway to mobilereceiver 120 data if mobile receivers 120 are not internet connecteddirectly. The central processing server 130 is preferably aninternet-connected general-purpose computer, but may additionally oralternatively comprise any suitable hardware.

2. RTK Satellite Positioning Method

As shown in FIG. 4, a method 200 for Real Time Kinematic (RTK) satellitepositioning includes: at a mobile receiver, receiving a navigationsatellite carrier signal S210, receiving a phase correction signal froma reference station S220, calculating integer phase ambiguity S230, andcalculating receiver position S240.

Step S210 includes receiving a navigation satellite carrier signal. StepS210 functions to provide the mobile receiver with a phase measurementand a pseudo-range measurement that can be used, along with a phasecorrection signal (received in Step S220) to calculate receiverposition. Navigation receiver carrier signals are preferably received atthe L1 frequency (1575.42 MHz), but may additionally or alternatively bereceived at the L2 frequency (1227.60 MHz) or any other suitablefrequency. Navigation satellite carrier signals received in Step S210may include GPS signals, GLONASS signals, Galileo signals, SBAS signalsand/or any other suitable navigation signal transmitted by a satellite.

Step S210 preferably includes receiving the navigation satellite carriersignal (which is an RF signal) at an RF antenna and converting thesignal to a digital baseband signal. This digital baseband signal ispreferably used for two tasks by Step S210: calculating the pseudo-rangefrom the receiver to the satellite (using standard GNSS time-of-flighttechniques) and measuring the relative phase of the carrier signal.

Step S210 is preferably performed for multiple satellites. The use ofpseudo-range and phase data from multiple satellites can provide formore accurate positioning, as described in later sections.

If receiver carrier signals are received at both L1 and L2 frequencies,Step S210 may include combining the L1 and L2 frequency signals for eachsatellite to create a beat signal. The resulting signal (i.e., the beatsignal) has a center frequency significantly lower than either the L1 orL2 signals (˜347.82 MHz), which allows for a smaller set of possibleinteger ambiguity values for a given prior (e.g., if |N|≦10 for an L1signal, |N|≦2 for the example beat signal). The resulting signal mayadditionally or alternatively possess other desirable properties (e.g.,reduction in ionospheric error).

In a variation of a preferred embodiment, the method 200 includes StepS211: transmitting carrier signal data (e.g., pseudo-range and/or phasedata) from the receiver to a remote computer (e.g., a computer at areference station, a cloud computing server). In this variation, StepsS220 through S240 may additionally be performed on the remote computer.

Step S220 includes receiving a phase correction (or phase observation)signal from a reference station. Step S220 functions to receive phasecorrection information used to determine, for a given satellite signal,the location of the mobile receiver. Step S220 preferably includesreceiving phase correction information for each satellite signalreceived in Step S210, but may additionally or alternatively includereceiving phase correction information for only a subset of thesatellite signals received in Step S210.

If Step S220 include receiving phase correction information for only asubset of satellite signals in Step S210, Step S220 may includeestimating phase correction information for any of the subset ofsatellite signals for which phase correction information is notreceived.

Step S220 includes receiving phase correction information from at leastone reference station, but may also include receiving phase correctioninformation from additional reference stations.

Step S220 may include receiving phase correction information for somesatellites from one reference station while receiving phase correctioninformation for other satellites from another reference station.Additionally or alternatively, Step S220 may include receiving phasecorrection information from multiple reference stations for a singlesatellite signal.

Step S220 preferably include receiving phase correction signals over aUHF radio (e.g., at 915 MHz), but may additionally or alternativelyinclude receiving phase correction signals over any suitablecommunication medium (e.g., an internet connection, a cellularconnection).

Phase correction signals preferably include carrier signal phase (asmeasured at the reference station) and reference station locationinformation (or other identifying information linked to location). Phasecorrection signals may additionally include pseudo-range data from thereference station, positioning code data, or any other relevant data.

Phase correction signals are preferably formatted as RTCMv3 messages,but may additionally or alternatively be formatted according to anysuitable standard or method. Reference stations used for transmittingphase correction signals may include dedicated RTK reference stations,Continuously Operating Reference Stations (CORS), Network RTK solutions(including virtual reference station solutions), or any other suitablereference station.

Step S230 includes calculating integer phase ambiguity. Step S230functions to allow for determination of the absolute difference in phasebetween a satellite carrier signal received at a reference station and asatellite carrier signal received at a mobile receiver, which in turnenables the position of the mobile receiver relative to the referencestation to be calculated.

Integer phase ambiguity is preferably calculated using doubledifferenced measurements of pseudo-range and relative phase.Double-differenced measurements are preferably calculated by taking thedifference of values for the difference of receiver and referencevalues. For example, the double-differenced measurements of pseudo rangeand phase for two satellites (satellite 1 and 2) can be modeled as

ρ₁₂=(ρ_(mr)−ρ_(ref))_(i=1)−(ρ_(mr)−ρ_(ref))_(i=2)

φ₁₂=(φ_(mr)−φ_(ref))_(i=1)−(φ_(mr)−φ_(ref))_(i=2)

where i is the satellite index, ρ_(mr),φ_(mr) are pseudo-range and phasemeasurements at the mobile receiver, and ρ_(ref),φ_(ref) arepseudo-range and phase measurements at the reference station.

More specifically, for a mobile receiver and a reference stationseparated by a vector b, the double differenced equations forpseudo-range ρ and phase φcan be written as

${\nabla{\Delta\rho}} = {\begin{pmatrix}\rho_{10} \\\vdots \\\rho_{n\; 0}\end{pmatrix} = {{{\begin{pmatrix}{e_{1} - e_{0}} \\\vdots \\{e_{n} - e_{0}}\end{pmatrix} \cdot b} + \varepsilon_{\rho}} = {{{DE} \cdot b} + \varepsilon_{\rho}}}}$${{\nabla\Delta}\; \varphi} = {\begin{pmatrix}\varphi_{10} \\\vdots \\\varphi_{n\; 0}\end{pmatrix} = {\frac{{DE} \cdot b}{\lambda} + N + \varepsilon_{\rho}}}$

where e_(n) is the unit line of sight vector to satellite n, ∈_(p)represents noise, λ is the wavelength of the carrier signal and N isinteger phase ambiguity. The use of double-differenced measurementsallows for the cancellation of satellite clock errors, receiver clockerrors, and some atmospheric error.

Step S230 preferably includes two substeps: generating a set ofhypotheses S231 and performing hypothesis testing on the set ofhypotheses S232. Additionally or alternatively, S230 may includecalculating integer phase ambiguity N using any number or type of steps.

Step S231 functions to produce a set of possible values for N as well asperform iterative refinement on that set. Step S231 preferably includesproducing a set of possible values for N using a Kalman filter process.

Kalman filters are recursive filters that estimate the state of a lineardynamic system based on a series of noisy measurements. In general form,the measurement equation appears as

z _(i) =H _(i) x _(i) +v _(i)

where z_(i) is the measurement at time (or step) i, x_(i) is the truestate, v_(i) is observation noise (zero mean and with known covariance),and H_(i) is the observation model that maps the true state space intothe observed space. The Kalman filter model further assumes that thereis a relationship between states at different times given by

x _(i) =F _(i) x _(i-1) w _(i)

where w_(i) is process noise (also zero mean and with known covariance)and F_(i) is the transition model that maps true state at time i−1 totrue state at time i.

In particular, Step S231 preferably includes producing a set of possiblevalues for N using a type of Kalman filter known as a Bierman-Thorntonfilter; additionally or alternatively, Step S231 may use any suitableprocess to produce possible values for N.

Starting with the equation

$\begin{pmatrix}{\nabla{\Delta\varphi}_{i}} \\{{\lambda {\nabla{\Delta\varphi}_{i}}} - {\nabla{\Delta\rho}_{i}}}\end{pmatrix} = {\begin{pmatrix}{\frac{1}{\lambda}{DE}_{i}} & I \\0 & {\lambda \; I}\end{pmatrix}\begin{pmatrix}b_{i} \\N\end{pmatrix}}$

and noting that

-   -   for any matrix A operating on a normally distributed random        variable x with covariance E, the random variable y=Ax will have        covariance AΣA^(T),    -   for the matrix A there are subspaces Ker[A] for which any        vectors x∈Ker[A] have the property 0=Ax        a matrix Q_(i) can be constructed such that 0=Q_(i)DE_(i) and        this matrix can be applied to form a second equation:

$\begin{pmatrix}{Q_{i}{\nabla{\Delta\varphi}_{i}}} \\{{\lambda {\nabla{\Delta\varphi}_{i}}} - {\nabla{\Delta\rho}_{i}}}\end{pmatrix} = {{\begin{pmatrix}{\frac{1}{\lambda}Q_{i}{DE}_{i}} & Q_{i} \\0 & {\lambda \; I}\end{pmatrix}\begin{pmatrix}b_{i} \\N\end{pmatrix}} = {{\begin{pmatrix}0 & Q_{i} \\0 & {\lambda \; I}\end{pmatrix}\begin{pmatrix}b_{i} \\N\end{pmatrix}} = {\begin{pmatrix}Q_{i} \\{\lambda \; I}\end{pmatrix}N}}}$

This equation relates phase change and pseudo-range directly to N(without inclusion of the baseline vector b). This equation can be usedas the measurement equation of the Kalman filter of Step S231 without acorresponding dynamic transition model. Calculating the value of Ndirectly (instead of attempting to calculate a Kalman filtered baseline)allows baseline states to be removed from the filter; because N isconstant, no dynamic transition model is needed. Removing therequirement for the dynamic transition model can substantially reducethe time and/or memory required to compute solutions, additionally,errors that might occur in a dynamic model cannot be explained away aserrors in estimates of N.

Computing Q_(i) requires knowledge of the line of sight vectorscontained in DE_(i). Step S231 preferably includes computing the line ofsight vectors from an estimate of b, (which, while not directlycalculated in previous calculations, can be found using a set of phasemeasurements and an estimate for N). Estimates of b are preferably foundas in Step S240, but may additionally or alternatively be found by anysuitable method. Additionally or alternatively, Step S231 may includecomputing the line of sight vectors from reference station data, or inany other suitable manner.

For a particular set of line of sight vectors, Q_(i) is preferablycomputed by generating a matrix whose rows form a basis for the leftnull space of DE, or Ker[DE^(T)]. This generation is preferably done viaQR decomposition, but may additionally or alternatively be performedusing singular value decomposition or any other suitable method.

From these equations, a set of hypotheses can be generated. Measurementsarising from ambiguity vector N are expected to be normally distributedand to have a mean given by the equation

$\begin{pmatrix}{Q_{i}{\nabla{\Delta\varphi}_{i}}} \\{{\lambda {\nabla{\Delta\varphi}_{i}}} - {\nabla{\Delta\rho}_{i}}}\end{pmatrix} = {\begin{pmatrix}Q_{i} \\{\lambda \; I}\end{pmatrix}N}$

The corresponding covariance is determined from the results of theKalman filter of Step S231, reducing the set of likely hypotheses (ascompared to covariance derived directly from the measurement model).From this information, a distribution of the set of hypotheses can befound.

Ideally, all hypotheses within a particular confidence interval aretested. The number of hypotheses contained within this interval(hereafter referred to as the testing set) is dependent on thecovariance for the N distribution. Since N needs to be computed forseveral satellites to determine position of the mobile receiver, thetotal set of hypotheses that need to be tested depends both on thecovariances for each satellite's associated N value, but also the numberof satellites.

Hypotheses are preferably bounded (for a particular confidence interval)by an ellipsoid defined by the covariance matrix. The ellipsoid definedby the covariance matrix is often extremely elongated, resulting a timeand computation intensive hypothesis generation process. To reduce thetime and computational resources required to perform this process, StepS231 may include performing a decorrelating reparameterization on thehypothesis search space, as shown in FIG. 5. Performing thisreparameterization transforms the hypothesis space such that theelongated ellipsoid is transformed to an approximate spheroid; thistransformation allows hypotheses to be identified substantially moreeasily. The hypotheses can then be transformed by an inversetransformation (inverse of the original reparameterization) to bereturned to the original coordinate space.

Step S231 preferably includes generating hypotheses for the testing setaccording to memory limits on the mobile receiver. For example, if areceiver features 64 kB of memory for storing hypotheses and storinginitial hypotheses for eight satellites requires 70 kB (while storinginitial hypotheses for 7 satellites requires only 50 kB), Step S231 mayinclude generating a set of initial hypotheses for 7 satellites, andthen adding hypotheses for the eighth satellite after enough of theinitial hypotheses for the 7 satellites have been eliminated. S231 mayadditionally or alternatively include waiting for the covariance of theKalman filter's estimate to shrink before generating hypotheses if thecurrent covariances are large enough that memory cannot store testingsets at some threshold confidence level for at least four satellites.

Though Step S231 is preferably performed before Step S232 is performedfor the first time, Step S231 may be performed again at any suitabletime to modify the set of hypotheses tested. For example, as theprobabilities for each set of hypotheses are refined by Step S232,hypotheses may be added to or subtracted from the testing set by StepS231. For example, if Step S231 produces a testing set A and later addsa testing set B containing new satellites, the new testing set may begenerated by taking the Cartesian/outer product of A and B, whereprobabilities are initialized via P(A,B)=P(A)/|B| where the denominatoris the number of hypotheses in set B and P(A) is the probability of Agenerated in Step S232. Step S232 may include initializing probabilitiesvia P(A,B)=P(A) (as Step S232 preferably tracks relative probabilitiesas opposed to absolute probabilities). If Step S231 includes dropping asatellite (e.g., if the satellite can no longer be tracked), this can beaccounted for by marginalizing hypotheses via P(A)=

P(A,B) over all hypotheses still in the set. Step S232 preferablyincludes tracking probabilities in log space; for l_(i)=ln [p_(i)]:

ln [p ₁ +p ₂]=ln [e ^(l) ¹ +e ^(l) ² ]=l ₁+ln [1+e ^(l) ¹ ^(-l) ² ]

Step S232 preferably includes approximating the logarithm term of l₁+ln[1+e^(l) ¹ ^(-l) ² ] via Taylor series in probability or log probabilityspace; additionally or alternatively, the logarithm term may beestimated as zero as the exponential term may be very small comparedto 1. This approximation may result in reducing computation time and/ormemory.

Step S232 functions to test the hypotheses of the refined set generatedby Step S231 in order to identify the hypothesis corresponding to theactual value of N. Step S231 preferably includes generating hypothesesusing LAMBDA or MLAMBDA algorithms using the means and covariancesgenerated by a Kalman filter, but may additionally or alternativelyinclude generating hypotheses using any other mean or covarianceestimate of N or according to any suitable algorithm. Step S232preferably includes testing hypotheses using a variation of thefollowing Bayesian update formula for a hypothesis h given anobservation y:

ln [P _(i)(h)]=l _(i)(h)=l _(i-1)(h)+ln [P(y _(i) |h)]n _(i)

where

$\eta_{i} = {\ln \left\lbrack {\sum\limits_{h \in \mathcal{H}}{{P\left( y_{i} \middle| h \right)}{P_{i - 1}(h)}}} \right\rbrack}$

Variables to be used in this equation are preferably defined accordingto the following definitions:

${r_{i} = {{\overset{\sim}{Q}}_{i}\begin{pmatrix}{\nabla{\Delta\varphi}_{i}} \\{\nabla{\Delta\rho}_{i}}\end{pmatrix}}};{{\overset{\sim}{Q}}_{i} = \begin{pmatrix}Q_{i} & 0 \\{\lambda \; I} & {- I}\end{pmatrix}}$

where r_(i) is distributed with mean and covariance

${{\overset{\_}{r}}_{iN} = {\begin{pmatrix}Q_{i} \\{\lambda \; I}\end{pmatrix}N}};{\Sigma_{i} = {{\overset{\sim}{Q}}_{i}{{Cov}\left\lbrack \begin{pmatrix}{\nabla{\Delta\varphi}_{i}} \\{\nabla{\Delta\rho}_{i}}\end{pmatrix} \right\rbrack}{\overset{\sim}{Q}}_{i}^{T}}}$

For observations y_(i)=r_(i) and hypotheses h=N, the previous hypothesisupdate formula can be written as

l _(i)(N)=l _(i-1)(N)−χ_(i) ²(N)+ln [k _(i) ]−n _(i);χ_(i) ²(N)=(r _(i)− r _(iN))^(T) Σi ⁻¹(r _(i) − r _(iN))

where k_(i) is the scaling factor in the normal distribution.

Step S232 preferably includes running the hypothesis test above untilthe ratio between the probabilities of the best two hypotheses reaches aset threshold, additionally or alternatively, Step S232 may includestopping the hypothesis test based on any other suitable condition(e.g., time).

Step S232 may additionally or alternatively include dropping hypothesesfrom further testing if their associated pseudo-likelihood, given by

l_(i)^(″)(N) = l_(i − 1)^(″)(N) − χ_(i)²(N) − l max_(i);${l\; \max_{i}} = {\max\limits_{N}\left\lbrack {{l_{i - 1}^{''}(N)} - {\chi_{i}^{2}(N)}} \right\rbrack}$

is less than some set threshold, the likelihood ratio test may beperformed in single precision for speed and numerical stability.Additionally or alternatively, Step S232 may include using any othersuitable metric for removing unlikely hypotheses; for example, removinghypotheses with a probability ratio (relative to the best hypothesis)below some threshold value.

Step S232 preferably includes calculating Σ_(i) and r_(i) only once perobservation step, as opposed to once per hypothesis N; additionally oralternatively, Step S232 may include calculating these at any suitabletime.

Step S240 includes calculating receiver position. Step S240 functions tocalculate the position of the mobile receiver based on the value for Ncomputed in Step S230. After N has been determined, the baseline vectorb for the mobile receiver is determined from the value(s) for N andphase/pseudo-range measurements by Step S240; this gives the position ofthe mobile receiver relative to a reference station. If the location ofthe reference station is known, Step S240 may include calculatingabsolute position of the mobile receiver (by applying b to the referencestation coordinates).

Step S240 may additionally include transmitting or storing receiverposition data. For instance, Step S240 may include transmitting receiverposition data from the receiver to an external computer over UHF radio,the internet, or any other suitable means.

All steps of the method 200 are preferably performed on a mobilereceiver, but additionally or alternatively, any step or set of stepsmay be performed on a remote platform (e.g., cloud computing servers ifthe mobile receiver has internet access).

The methods of the preferred embodiment and variations thereof can beembodied and/or implemented at least in part as a machine configured toreceive a computer-readable medium storing computer-readableinstructions. The instructions are preferably executed bycomputer-executable components preferably integrated with an RTK-capablemobile GNSS receiver. The computer-readable medium can be stored on anysuitable computer-readable media such as RAMs, ROMs, flash memory,EEPROMs, optical devices (CD or DVD), hard drives, floppy drives, or anysuitable device. The computer-executable component is preferably ageneral or application specific processor, but any suitable dedicatedhardware or hardware/firmware combination device can alternatively oradditionally execute the instructions.

As a person skilled in the art will recognize from the previous detaileddescription and from the figures and claims, modifications and changescan be made to the preferred embodiments of the invention withoutdeparting from the scope of this invention defined in the followingclaims.

We claim:
 1. A method for Real Time Kinematic satellite positioningcomprising: at a mobile receiver, receiving a first navigation satellitecarrier signal from a first navigation satellite, receiving a secondnavigation satellite carrier signal from a second navigation satellite,receiving a third navigation satellite carrier signal from a thirdnavigation satellite, and receiving a fourth navigation satellitecarrier signal from a fourth navigation satellite; at the mobilereceiver, receiving first, second, third, and fourth phase correctionsignals from a reference station; at the mobile receiver, calculating aset of integer phase ambiguities from double-differenced measurements ofpseudo-range and phase; and at the mobile receiver, calculating arelative position of the mobile receiver from the set of integer phaseambiguities and the double-differenced measurements of pseudo-range andphase.
 2. The method of claim 1, wherein receiving a navigationsatellite carrier signal comprises receiving a first GPS signal on an L1frequency.
 3. The method of claim 2, wherein receiving a navigationsatellite carrier signal further comprises receiving a second GPS signalon an L2 frequency and combining the first and second GPS signals tocreate a beat signal.
 4. The method of claim 1, further comprisingreceiving a set of additional phase correction signals from a secondreference station; wherein calculating the relative position of themobile receiver comprises calculating the relative position of themobile receiver at least in part on the set of additional phasecorrection signals.
 5. The method of claim 1, wherein receiving thefirst, second, third, and fourth phase correction signals comprisesreceiving the first, second, third, and fourth phase correction signalsfrom a UHF radio.
 6. The method of claim 1, wherein calculating a set ofinteger phase ambiguities comprises generating a first set of integerphase ambiguity hypotheses using a Kalman filter and performinghypothesis testing on the first set of integer phase ambiguityhypotheses.
 7. The method of claim 6, wherein generating the first setof integer phase ambiguity hypotheses comprises generating the first setof integer phase ambiguity hypotheses using means and covariancesgenerated by a Bierman-Thornton filter and at least one of a LAMBDA andan MLAMBDA algorithm.
 8. The method of claim 7, wherein generating thefirst set of integer phase ambiguity hypotheses comprises generating thefirst set of integer phase ambiguity hypotheses from a measurementequation that relates phase change and pseudo-range to integer ambiguitywithout inclusion of a baseline vector; wherein generating the first setof integer phase ambiguity hypotheses comprises generating the first setof integer phase ambiguity hypotheses without a dynamic transitionmodel.
 9. The method of claim 8, wherein generating the first set ofinteger phase ambiguity hypotheses further comprises performing adecorrelating reparameterization of a hypothesis search space.
 10. Themethod of claim 9, wherein performing hypothesis testing comprisesperforming hypothesis testing using a Bayesian update algorithm.
 11. Themethod of claim 10, wherein performing hypothesis testing comprisesperforming hypothesis testing comprises removing a hypothesis fromfurther testing based on a pseudo-likelihood of the hypothesis passing athreshold value.
 12. The method of claim 11, wherein performinghypothesis testing comprises ceasing hypothesis testing when a ratio ofprobability of a most likely hypothesis to probability of a second-mostlikely hypothesis passes a threshold value.
 13. The method of claim 12,further comprising generating a second set of integer phase ambiguityhypotheses after hypotheses from the first set of integer phaseambiguity hypotheses have been removed from further testing.
 14. Themethod of claim 13, wherein generating the second set of integer phaseambiguity hypotheses comprises generating the second set of integerphase ambiguity hypotheses in response to a hypothesis search spacebecoming smaller than a threshold value.
 15. The method of claim 14,wherein calculating the set of integer phase ambiguities comprisestracking relative probabilities in logarithmic space.
 16. A method forReal Time Kinematic satellite positioning comprising: at a mobilereceiver, receiving a first navigation satellite carrier signal from afirst navigation satellite, receiving a second navigation satellitecarrier signal from a second navigation satellite, receiving a thirdnavigation satellite carrier signal from a third navigation satellite,and receiving a fourth navigation satellite carrier signal from a fourthnavigation satellite; at the mobile receiver, transmitting carriersignal data corresponding to the first, second, and third navigationsatellites from the mobile receiver to a remote computer; at the remotecomputer, receiving first, second, third, and fourth phase correctionsignals from a reference station; at the remote computer, calculating aset of integer phase ambiguities from double-differenced measurements ofpseudo-range and phase; and at the remote computer, calculating arelative position of the mobile receiver from the set of integer phaseambiguities and the double-differenced measurements of pseudo-range andphase.
 17. The method of claim 16, wherein calculating a set of integerphase ambiguities comprises: generating a first set of integer phaseambiguity hypotheses using means and covariances generated by aBierman-Thornton filter and at least one of a LAMBDA and an MLAMBDAalgorithm; and performing hypothesis testing on the first set of integerphase ambiguity hypotheses.
 18. The method of claim 17, whereingenerating the first set of integer phase ambiguity hypotheses comprisesgenerating the first set of integer phase ambiguity hypotheses from ameasurement equation that relates phase change and pseudo-range tointeger ambiguity without inclusion of a baseline vector; whereingenerating the first set of integer phase ambiguity hypotheses comprisesgenerating the first set of integer phase ambiguity hypotheses without adynamic transition model.
 19. The method of claim 18, wherein generatingthe first set of integer phase ambiguity hypotheses further comprisesperforming a decorrelating reparameterization of a hypothesis searchspace.
 20. The method of claim 19, wherein performing hypothesis testingcomprises performing hypothesis testing using a Bayesian updatealgorithm.